Unlimited sheds or trackers 2D (PVsyst)

Method

The rows are treated as infinitely long and the geometry is solved in the plane across them. Each row is its total width, inactive bands included. A fixed row rises at the tilt from a bottom edge at the height above ground of the System tab. A tracker row turns about an axis at that height, on a level axis. The ground of one pitch is cut into 100 points. Each point receives the uniform part of the sky light in proportion to the sky it sees past the rows (1). It also receives the direct and circumsolar light on the share of its ground cell outside the shadow of a row, and the transparent fraction of the module plane times that light on the shaded share (2). The ground reflects the albedo fraction of its light evenly in every direction. From each point, 180 rays spread over the half circle above it. Each ray counts for the first face it meets, the rear face of one row or the front face of another, weighted by the transmission of plain glass at the angle it arrives (3, 4). A ray that meets an inactive band is lost. The rear side also receives the sky light from behind the plane (5, 6) and, on fixed rows, the direct sunlight when the sun is behind it (7). The light on the front faces is the Ground reflection on front side row of the loss diagram. The structure shades a fixed fraction of the rear side (8), and the module converts the front irradiance plus the rear irradiance scaled by the bifaciality factor (9).

On trackers the sky view of each ground point, its two ray sums and the rear sky factor are computed for 7 rotations evenly spread between the rotation limits, and each timestep reads them from a natural cubic spline in its rotation (10). The ground points stay fixed under the tracker axis, and the shadow on the ground is the timestep's own. A tracker's rear side receives no direct sunlight.

The scales $k_{bck}$ and $k_{frt}$ are fitted constants on the ray sums of (4), which leave out the cosine at the receiving face. The tilt law of the rear sky factor (5) is fitted too. Both hold for fixed rows and trackers.

Settings

Setting	Value
Ground points across one pitch	100
Rays from each ground point	180, over the half circle above it
Rows traced for the sky of a ground point	30 on each side
Rows traced for the rays	50 on each side
Scale of the ground light on the rear side $k_{bck}$	0.9733
Scale of the ground light on the front side $k_{frt}$	0.9357
Tilt law of the rear sky factor	(β − 5.51°)/78.43°, held between 0 and 1
Tracker rotations computed	7, evenly spread between the rotation limits
Largest tracker axis tilt	2°

The math

1. The sky each ground point sees
2. The light on each ground point
3. The share of the reflected light that reaches a face
4. The ground light on the two faces
5. The sky light behind the plane
6. The sky the rear side sees past the next row
7. The direct sunlight on the rear side
8. The light on the rear side
9. The light the module converts
10. A tracker's integrals at its rotation

$$F_x = 1 - \frac{1}{2}\sum_{n}\left(\sin a_{n,2} - \sin a_{n,1}\right) \tag{1}$$ $$G(x) = D_{iso,h}\,F_x + (B_h + C_h)\,\big[s_x + f_{tr}\,(1 - s_x)\big] \tag{2}$$ $$T_f(x) = \sum_{j \to f} \tfrac{1}{2}\cos\phi_j\,\Delta\phi\ F_g(\theta_j),\qquad \Delta\phi = \frac{\pi}{180} \tag{3}$$ $$G_{refl} = k_{bck}\,\text{albedo}\,\frac{P}{W}\,\big\langle T_{rear}(x)\,G(x) \big\rangle,\qquad G_{frt} = k_{frt}\,\text{albedo}\,\frac{P}{W}\,\big\langle T_{front}(x)\,G(x) \big\rangle \tag{4}$$ $$G_{sky,rear} = K(\beta)\,\big(D_{iso,r} + D_{hb,r}\big),\qquad K(\beta) = M(\beta)\,\min\!\Big(1,\ \max\!\Big(0,\ \frac{\beta - 5.51°}{78.43°}\Big)\Big) \tag{5}$$ $$M(\beta) = \frac{1}{200}\sum_{j=1}^{200} \frac{1 - \cos(\beta - e_j)}{1 - \cos\beta} \tag{6}$$ $$G_{beam,rear} = \Big(\text{DNI} + \frac{D\,F_1}{\sin\alpha}\Big)\max(0,\ -\cos\theta)\,F_g(180° - \theta)\,s_{lit} \tag{7}$$ $$G_{rear} = (1 - f_{struct})\,\big(G_{refl} + G_{sky,rear} + G_{beam,rear}\big) \tag{8}$$ $$G = G_{front} + b_f\,G_{rear} \tag{9}$$ $$Q(\rho) = \sum_{k=1}^{7} w_k(\rho)\,Q(\rho_k),\qquad \rho_k = \rho_{min} + (k - 1)\,\frac{\rho_{max} - \rho_{min}}{6} \tag{10}$$

where

Symbol	Meaning	Unit
$F_x$	the sky view factor of the ground point at $x$	—
$n$	the rows within 30 pitches on either side of the point	—
$a_{n,1},\ a_{n,2}$	the angles from the vertical, at the point, to the two edges of row $n$, the smaller one first, with the ranges of overlapping rows merged and held within ±90°	degrees
$G(x)$	the irradiance on the ground at $x$	W/m²
$D_{iso,h}$	the uniform part of the diffuse light on a horizontal plane, from the transposition	W/m²
$B_h$	the direct light on a horizontal plane: the global minus the diffuse horizontal irradiance	W/m²
$C_h$	the circumsolar part of the diffuse light on a horizontal plane, from the transposition	W/m²
$f_{tr}$	the transparent fraction of the module plane, System tab	—
$s_x$	the share of the ground cell around $x$, one hundredth of the pitch wide, outside the shadow of the rows' total width	—
$T_f(x)$	the share of the light reflected at $x$ that reaches face $f$ through its glass: $f$ is the rear face of the row or the front face of the next row	—
$j \to f$	the rays from $x$ whose first face met is $f$, on its sensitive width	—
$\phi_j$	the angle of ray $j$ from the vertical	degrees
$\Delta\phi$	the angle between two rays	rad
$\theta_j$	the angle between ray $j$ and the perpendicular to the face it meets	degrees
$F_g$	the transmission of plain glass (refractive index 1.526) at the angle in brackets, relative to its transmission at normal incidence	—
$G_{refl}$	the ground-reflected light on the rear side	W/m²
$G_{frt}$	the ground-reflected light on the front face of the next row	W/m²
$k_{bck},\ k_{frt}$	the scales of the ground light on the rear and the front side	—
$\text{albedo}$	the ground albedo: the bifacial albedo when it is set, otherwise the site's albedo of the month	—
$P$	the row pitch, the distance between the rows	m
$W$	the collector width	m
$\langle \cdot \rangle$	the mean over the 100 ground points	—
$G_{sky,rear}$	the sky light on the rear side	W/m²
$K(\beta)$	the rear sky factor at tilt $\beta$	—
$\beta$	the tilt of the module plane	degrees
$D_{iso,r},\ D_{hb,r}$	the uniform and horizon-band parts of the transposition on a plane facing the rear, tilted 180° minus the module tilt	W/m²
$M(\beta)$	the rear side's view of the sky left by the next row, over its view without that row	—
$e_j$	the elevation of the next row's top edge seen from point $j$ of 200 across the sensitive width of the rear face, 0 when it lies below the horizon	degrees
$G_{beam,rear}$	the direct sunlight on the rear side	W/m²
$\text{DNI}$	the direct normal irradiance of the weather file	W/m²
$D$	the diffuse horizontal irradiance of the weather file	W/m²
$F_1$	the share of the diffuse light the transposition puts in the circumsolar part, $1 - D_{iso,h}/D$	—
$\alpha$	the elevation of the sun above the horizon	degrees
$\theta$	the angle between the sun and the perpendicular to the module plane	degrees
$s_{lit}$	the fraction of the rear side the next row leaves lit	—
$G_{rear}$	the irradiance on the rear side	W/m²
$f_{struct}$	the fraction of the rear side shaded by the structure	—
$G$	the irradiance the module model converts	W/m²
$G_{front}$	the effective irradiance on the front side after the shading, soiling and glass-reflection steps, plus $G_{frt}$	W/m²
$b_f$	the bifaciality factor of the module: its rear-side efficiency relative to the front	—
$Q$	a tracker integral: a ground point's sky view $F_x$, its ray sums $T_f(x)$, or the rear sky factor $K$	—
$\rho$	the tracker rotation of the timestep	degrees
$\rho_k$	the 7 rotations the integrals are computed at	degrees
$\rho_{min},\ \rho_{max}$	the rotation limits	degrees
$w_k(\rho)$	the weight of $\rho_k$ in the natural cubic spline through the 7 rotations, read at $\rho$	—

References

1. Marion et al. 2017, A practical irradiance model for bifacial PV modules, NREL/CP-5J00-67847
2. PVsyst 8 help, Bifacial systems, 2D unlimited sheds model
3. PVsyst 8 help, Bifacial systems, 2D unlimited trackers model
4. Analysis of PVsyst 8.1.5 outputs shows this convention in its results.

API

bifacial.model = sheds_2d

Symbols follow the symbol registry.